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Physical review. B, Condensed matter

Single-electron oscillations in the one-dimensional interacting electron gas

D. V. Averin, Yuli V. Nazarov

Published Jan 15, 1994• 3 citations
Publisher

Abstract

We have shown that the one-dimensional interacting electron gas carrying dc current I should exhibit single-electron singularities at frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$=2\ensuremath{\pi}I/e for arbitrary strength of electron-electron interaction. These singularities arise due to Coulomb or Fermi correlations between electrons in the gas. Depending on the strength of the interaction the frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{s}}$ can be interpreted either as a frequency of the single-electron oscillations or the cutoff frequency of the shot noise associated with the current I.

DOI
10.1103/physrevb.49.2951

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We have shown that the one-dimensional interacting electron gas carrying dc current I should exhibit single-electron singularities at frequency \ensuremathωs{\mathrm{\ensuremath{\omega}}}_{\mathit{s}}\ensuremathωs​=2\ensuremath{\pi}I/e for arbitrary strength of electron-electron interaction. These singularities arise due to Coulomb or Fermi correlations between electrons in the gas. Depending on the strength of the interaction the frequency \ensuremathωs{\mathrm{\ensuremath{\omega}}}_{\mathit{s}}\ensuremathωs​ can be interpreted either as a frequency of the single-electron oscillations or the cutoff frequency of the shot noise associated with the current I.
The photon-absorption edge in a weakly interacting one-dimensional electron gas is studied, treating backscattering of conduction electrons from the core hole exactly. Close to threshold, there is a power-law singularity in the absorption, I(\ensuremath{\epsilon})\ensuremath{\propto}${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$, with \ensuremath{\alpha}=3/8+${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$, where ${\mathrm{\ensuremath{\delta}}}_{+}$ is the forward-scattering phase shift of the core hole. In contrast to previous theories, \ensuremath{\alpha} is finite (and universal) in the limit of weak core-hole potential. In the case of weak backscattering U(2kF{\mathit{k}}_{\mathit{F}}kF​), the exponent in the power-law dependence of absorption on energy crosses over to a value \ensuremath{\alpha}=${\mathrm{\ensuremath{\delta}}}_{+}$/\ensuremath{\pi}-${\mathrm{\ensuremath{\delta}}}_{+}^{2}$/2${\mathrm{\ensuremath{\pi}}}^{2}$ above an energy scale ${\mathrm{\ensuremath{\epsilon}}}^{\mathrm{*}}$\ensuremath{\sim}[U(2kF{\mathit{k}}_{\mathit{F}}kF​)${]}^{1/\ensuremath{\gamma}}$, where \ensuremath{\gamma} is a dimensionless measure of the electron-electron interactions.
We compute the single-particle spectral density, susceptibility near the Kohn anomaly, and pair propagator for a one-dimensional interacting-electron gas. With an attractive interaction, the pair propagator is divergent in the zero-temperature limit and the Kohn singularity is removed. For repulsive interactions, the Kohn singularity is stronger than the free-particle case and the pair propagator is finite. The low-temperature behavior of the interacting system is not consistent with the usual Ginzburg-Landau functional because the frequency, temperature, and momentum dependences are characterized by power-law behavior with the exponent dependent on the interaction strength. Similarly, the energy dependence of the single-particle spectral density obeys a power law whose exponent depends on the interaction and exhibits no quasiparticle character. Our calculations are exact for the Luttinger or Tomonaga model of the one-dimensional interacting system.
We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency ${\ensuremath{\omega}}_{D}$, while the electron bandwidth is of order EF{E}_{F}EF​, with ${\ensuremath{\omega}}_{D}<{E}_{F}$. Our results, consistent only in the region of weak coupling, give the scaling trajectories for the coupling constants and cutoffs. The relationship to the conventional one-cutoff renormalization-group approach is discussed.
The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.